Proposition 1

Let E be a locally convex space. Every finite-dimensional vector subspace of E is direct.

Let F be a finite-dimensional vector subspace of E. Let S denote a topological supplement of the closure of $\{0\}$ in F, which is a direct subspace of E. The space $F_2= F\cap S$ is separated and finite-dimensional; there therefore exists an integer $n\geqslant 0$ such that $F_2$ is isomorphic to $K^n$ (TVS, I, p. 14, th. 2). There then exists a continuous linear mapping $p: S\rightarrow F_2$ extending the identity mapping of $F_2$ (TVS, II, p. 26, remark). The kernel of $p$ is a topological supplement of $F_2$ in S, and also of F in E.